It's not surprising that this is a confusing thing, as prior to 4.0 dbms ~ database ~ graph, since you could really only have 1. Multi-database functionality is forcing us to clean up and clarify our language here. You may be using an older version of Neo4j Desktop (you may need to download a newer install), as the latest versions of Desktop now use "Add Database" instead of "Add Graph"...but unfortunately this still isn't quite accurate.
What you're adding in Desktop is more like a dbms, a Neo4j instance.
Once you start that up, it will be default have 2 databases (neo4j and system) and you can of course add more.
Currently we have 1 graph per database, so at this time database ~ graph, but we are looking into multi-graph features, allowing multiple graphs per db, so these two terms will no longer be interchangeable at some point.
So the tl;dr is:
A dbms is a container for 1 or more databases (in 4.0 you will have 2 by default, due to the system db)
A database is a container for 1 or more graphs (though multiple are not currently possible)
Work on clarifying and cleaning up our terms accordingly is ongoing.